A few Mandelbrot variations I discovered (Stay tuned for more!)

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Offline greentexas

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« on: September 06, 2017, 05:17:01 PM »
This is a set of ten Mandelbrot variations I discovered. Some of them may have already been discovered, some of them have no Minibrots, but stay tuned for more fractals.

VARIATION 1
ozr = zr;
zr = zr*zr - zi*abs(zi) + JuliaR;
zi = 2*ozr*zi + JuliaI;

VARIATION 2
ozr = zr;
zr = zr*zr - zi*zi + JuliaR;
zi = 2*ozr*zi*zi + JuliaI;

VARIATION 3
if (zr > 0) {
ozr = zr;
zr = zr*zr - zi*zi + JuliaR;
zi = 2*ozr*abs(zi) + JuliaI;
} else {
ozr = zr;
zr = zr*zr - zi*zi + JuliaR;
zi = 2*abs(ozr)*zi + JuliaI;
}

VARIATION 4
ozr = zr;
zr = zr*zr - zi*zi - 2*zr*zi + JuliaR;
zi = 2*ozr*zi - 2*zr*zi*zi + JuliaI;

VARIATION 5
ozr = zr;
zr = zr*zr*zr - 1.5*zi*zi*zi + JuliaR;
zi = -3*ozr*ozr*zi + JuliaI;

VARIATION 6
ozr = zr;
zr = (zr*zr - zi*zi)*(zr*zr - zi*zi) + JuliaR;
zi = 4*ozr*ozr*zi*zi + JuliaI;


VARIATION 7

ozr = zr;
zr = (zr*zr - zi*zi)*(zr - zi) + JuliaR;
zi = 3*ozr*zi + JuliaI;

VARIATION 8

if (zi > 0) {
ozr = zr;
zr = abs(zr*zr - zi*zi) + JuliaR;
zi = 2*ozr*zi + JuliaI;
} else {
ozr = zr;
zr = (zr*zr - zi*zi) + JuliaR;
zi = 2*ozr*zi + abs(JuliaI);   
}

VARIATION 9

ozr = zr;
zr = (zr*zr - zi*zi) + JuliaR;
zi = -2*abs(ozr)*zi + JuliaI;
ozr = zr;
zr = (zr*zr - zi*zi);
zi = 2*abs(ozr)*zi + JuliaI;

VARIATION 10

ozr = zr;
zr = (zr*zr - zi*zi) + JuliaR;
zi = 2*abs(ozr)*zi + JuliaI;
ozr = zr;

zr = (zr*zr - zi*zi) + JuliaR;
zi = 2*(ozr)*zi + JuliaI;

Offline RedshiftRider

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« Reply #1 on: September 06, 2017, 09:28:07 PM »
Hey Greentexas, good to see you found your way here. Welcome!


Great idea I am curious how they look, could you provide a visual representation of the the different variants?

Offline Softology

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« Reply #2 on: September 07, 2017, 12:19:01 AM »
Assuming I didn't make any mistakes with the formulas, here are some sample images and a zoomed in preview for each of the first 9.

Variation 1





Variation 2





Variation 3





Variation 4





Variation 5





Variation 6





Variation 7





Variation 8





Variation 9





Variation 9 is the Heart Mandelbrot from stardust4ever's variations



Jason.

Offline greentexas

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« Reply #3 on: September 07, 2017, 04:16:41 AM »
Thank you so much! I wouldn't think anyone would be so kind to do that for me! When I created formula #9, it did not look like the Heart. I tried to make sure it wasn't like his 4th Quasi Heart either. Stardust4ever's fractals were an inspiration for me to make this.

Offline RedshiftRider

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« Reply #4 on: September 07, 2017, 04:00:30 PM »
These look quite interesting.


I think I have seen variation 3 before. Isn't that some sort of combination (well all of them probably are) of the m-set and the burning ship?

Offline greentexas

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« Reply #5 on: September 09, 2017, 03:16:48 AM »
Variation 3 is not a Mandelbrot/Burning Ship hybrid like stardust4ever's fractals, but 50% of the numbers used are Mandelbrot/Mandelbar, and the other 50% is Burning Ship.

Let's say we have this psuedocode:
if zr > 0 {
do(Mandelbrot)
} else {
do(Mandelbar)
}

It creates the Perpendicular Mandelbrot.

This formula's psuedocode would be:
if zr > 0 {
do(Perpendicular Burning Ship)
} else {
do(Perpendicular Mandelbrot)
}

making it like a hybrid of a hybrid. There is one Mandelbar, one Mandelbrot, and two Burning Ships.

Offline greentexas

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« Reply #6 on: September 09, 2017, 03:37:45 AM »
Ten more variations! (The first time I tried plotting the Cubic Mandelbrot using Scratch a couple years ago, I think version 15 was a mistake I made. I was pretty silly back then, to be using ABS.)

Variation 20 is a 6th order variation of variation 1.

I'll send an image soon of the ten variations.

VARIATION 11
ozr = zr;
zr = zr*zr*(1 + zi) - (zi*zi * (1 + zr)) + JuliaR;
zi = -2*ozr*zi + JuliaI;

VARIATION 12
ozr = zr;
zr = -(zr*zr - 3*zi*zi)*(zr) + JuliaR;
zi = -abs(3*ozr*ozr - zi*zi)*(zi) + JuliaI;

VARIATION 13
ozr = zr;
zr = (zr*zr*zr - 3*zi*zi)*(zr) + JuliaR;
zi = (3*ozr*ozr*zi - zi*zi)*(zi) + JuliaI;

VARIATION 14
ozr = zr;
zr = (zr*zr - 3*zr*zi*zi) + JuliaR;
zi = (3*ozr*ozr - zi*zi)*(zi) + JuliaI;

VARIATION 15
ozr = zr;
zr = (zr*zr*zr - 3*zr*zi*zi) + JuliaR;
zi = (3*abs(ozr*ozr)*zi - abs(zi)*zi*zi) + JuliaI;

VARIATION 16
ozr = zr;
zr = (zr*zr*zr - 3*abs(zr)*zi*zi) + JuliaR;
zi = (3*abs(ozr*ozr)*zi - (zi)*zi*zi) + JuliaI;

VARIATION 17
ozr = zr;
zr = abs(zr*zr*abs(zr)- 3*abs(zr)*zi*abs(zi)) + JuliaR;
zi = abs(3*abs(ozr*ozr)*zi - abs(zi)*zi*zi) + JuliaI;

VARIATION 18
ozr = zr;
zr = (zr*zr*(zr)-(zi*zr) + 3*(zr)*zi*(-zi)) + JuliaR;
zi = (3*(ozr*ozr*ozr)*zi - (zi)*zi*zi) + JuliaI;

VARIATION 19
ozr = zr;
zr = 2*zr*zi + JuliaI;
zi = ozr*ozr*(ozr + zi) - zi*zi + JuliaR;

VARIATION 20
ozr = zr;
zr = zr*zr - zi*zi*zi;
zi = 2*ozr*zi;
ozr = zr;
zr = zr*zr - zi*zi*zi + JuliaR;
zi = 2*ozr*zi + JuliaI;
« Last Edit: September 11, 2017, 11:11:40 PM by greentexas »

Offline Softology

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« Reply #7 on: September 14, 2017, 12:03:26 AM »
OK, I gave those ones a go too.  A lot of them are distorted stretched "whipped cream" results that are not too interesting, but a few are OK.

Variation 11





Variation 12





Variation 13





Variation 14





Variation 15





Variation 16





Variation 17





Variation 18





Variation 19





Variation 20





All of those were rendered in http://softology.com.au/voc.htm.

Jason.

Offline greentexas

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« Reply #8 on: October 03, 2017, 11:11:54 PM »
Here are ten more variations. The last ten weren't very good, I'll admit it. I discovered variation 29 when I was about nine, and nicknamed it the "sitting cheese curl".

VARIATION 21
ozx = zx;
zx = zx*zx - zy*zy*(3 + 4*zy) + cx;
zy = 2*ozx*zy + cy;

VARIATION 22
ozx = zx;
zx = abs(zx - 0.125*zy);
zy = zy + 0.0625*zx*zx;
zx = zx;
zx = zx*zx - zy*zy + cx;
zy = 2*ozx*zy + cy;

VARIATION 23
ozx = zx;
zx = zx*abs(zx) - zy*zy;
zy = 2*ozx*(zy);
ozx = zx;
zx = zx*zx - zy*zy + cx;
zy = 2*ozx*abs(zy) + cy;

VARIATION 24
ozx = zx;
zx = zx*zx + zy*zy - zy*zy*zy + cx;
zy = -2*ozx*zy + (cy);


VARIATION 25

ozx = zx;
zx = 3*zx*abs(zx)*zy - zy*zy*zy + cx;
zy = -abs(ozx)*ozx*ozx - 3*ozx*abs(zy)*zy + cy;


VARIATION 26

ozx = zx;
zx = zx*zx - zy*zy + cx;
zy = 2*ozx*zy + (cy);
if (zx > zx*sqrt(sqrt(abs(zx)))) {
zy = -zy;  
}


VARIATION 27

ozx = zx;
zx = sqrt(zx*zx + zy*zy)*(zx*zx - zy*zy) + cx;
zy = (-2*ozx*zy)*sqrt(abs(ozx*zy)) - cy;


VARIATION 28

ozx = zx;
zx = (zx*zx - zy*zy) - cx;
zy = (2*ozx*zy) - cy;
ozx = zx;
zx = (zx*zx - zy*zy) - cx;
zy = (2*ozx*zy) + cy;
ozx = zx;
zx = (zx*zx - zy*zy) + cx;
zy = (2*ozx*zy) - cy;
ozx = zx;
zx = (zx*zx - zy*zy) + cx;
zy = (2*ozx*zy) + cy;


VARIATION 29

ozx = zx;
zx = (zx*zx - zy*zy) + zy + cx;
zy = (2*ozx*zy) - zy - cy;


VARIATION 30

ozx = zx;
zx = (zx*zx - zy*zy) + cx;
zy = (2*ozx*zy) + cy;
if (zx - zy * zx > zy) {
zx = zx-zy*zy;   
}

Offline Fraktalist

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« Reply #9 on: October 04, 2017, 12:11:23 AM »
very interesting!

there was this fascinating video over at ff1.com, where these variations were shown as a complete morph-cycle, making it visually obvious how the burning ship (if I remember  corectly) was just a variation of the mset..

I couldn'r find it - if anyone does remember the video, please post here!

Offline Softology

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« Reply #10 on: October 04, 2017, 04:36:55 AM »
Here are ten more variations. The last ten weren't very good, I'll admit it. I discovered variation 29 when I was about nine, and nicknamed it the "sitting cheese curl".

It would be really helpful if you included your own sample pics with formulas.  At least then I know if I have the formula right when I convert them into GLSL for my program.

Offline F. Bernkastel

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« Reply #11 on: December 15, 2017, 10:17:25 AM »
there was this fascinating video over at ff1.com, where these variations were shown as a complete morph-cycle, making it visually obvious how the burning ship (if I remember  corectly) was just a variation of the mset..

Do you mean something like this?
\( z_{n+1} = \left\{\begin{matrix}
\|z_{n}\| > \psi,   & z_{n}^{2} + \textrm{C}\\
\|z_{n}\| \leq  \psi,   & ( |\Re{(z_{n}})| + i\,|\Im{(z_{n}})|)^{2} + \textrm{C}
\end{matrix}\right. \)

Where Ψ is a random value with normal distribution. from 0 to 2.
« Last Edit: February 21, 2018, 08:15:15 AM by F. Bernkastel »
...Then clasp me round the neck once more, and give me one more kiss;
And now mine own dear little girl, there is no way but this.

Offline Kosmic Stardust

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« Reply #12 on: November 01, 2018, 10:31:43 AM »
Once again, forgive me for the necrobumps. :fp:

I came across a bunch of "whipped cream" type renders when generating my fractal formulas. One thing that helps is to look at how the polynomial breakdown for complex math plays out. The distributive property groups the polynomials into logical building blocks. The distributive property not only simplifies the strings visually but reduces the cpu complexity, especially if R^2 and I^2 are factored out.

Typically there is a group of terms in the parenthesis followed by a multiplicand of I or R component. Applying the abs() function and or a sign change to these groups individually creates all of the variations of each set. These are interesting fractals in the sense that they can be explored to great depths with minis and minimal distortion. Normally with two manipulatable terms in the real and imaginary components of the formula, this leads to 12 variations for each power.

4th order gets 24 formula instead of 12 due to the symmetry of the fractal resulting in more variations without flips or mirrors. 5th order results in a 4-sided fractal, with additional flips and rotations of existing abs () formula which don't create unique fractals as in 4th. Each fractal has 64 possible outcomes for optionally applying a sign change or abs to each term. Eiminating flips and rotations yeilds 12 possible outcomes, 24 for 4th order. Half of those 64 combinations are asymmetric fractals which exist as chiral pairs. Eliminating mirrors of the asymmetric fractals (example: burning ship with mast above and below the plane) always yields multiples of 3 possible fractal combinations.

When you factor out all of the individual terms in the polynomial, you increase the render time and number of necessary multiplications. You also get the "whipped cream" fractals whenever abs() of sign change is applied individually to one of the terms. I am sure these fractals have depth but the distortion effects are ever changing so one would have to continually adjust the aspect ratio and skew angle throughout the zoom.

The "heart" fractal belongs in the perpendicular family and contains the bottom half of the burning ship flipped about the x axis. Other heart variants contain other fractal's lesser halfs flipped across the axis as well. It is like a deflated Mandelbrot, containing only minis on the needle. KF chose not to add these to his software because they were "uninteresting", a conclusion I made as well.

Good luck with your variations. I believe there are many more interesting "mixed power" polynomial fractals to discover, like the HPDZ Buffalo (first and second order polynomial hybrid). Interestingly the HPDZ buffalo required a non-zero seed to produce minis. Starting with zero looked the same at a glance but you could zoom forever without striking a mini. The HPDZ Buffalo fits the ax^2 + bx^2 + c quadratic mold. There are likely a crapton of interesting polynomial fractals just like this.

Ultrafractal also had a plugin that made some interesting cases. One of them was a hybrid third order formula resembled a 4-bulb Mandelbrot with no elephant valley region, and the Seahorse Valleys had different periods compared to the classic set. Then there is the lamda, a two bulbed Mandelbrot, and others. It had severe limitations however, like complete lack of 64-bit support and render speed went to dinosaur slow once you hit arbitrary.
« Last Edit: November 01, 2018, 10:42:05 AM by Kosmic Stardust »

Offline v

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« Reply #13 on: November 02, 2018, 11:15:12 AM »
I did a continuous morph of the function y=x to y=|x| as it applies in the mset and burning ship

https://fractalforums.org/index.php?action=gallery;sa=view;id=414


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